Let $K$ be a complete discrete valuation field of characteristic $(0, p)$ with perfect residue field, and let $T$ be an integral $\mathbb{Z}_p$-representation of $\mathrm{Gal}(\overline{K}/K)$. A theorem of T. Liu says that if $T/p^n T$ is torsion semi-stable (resp. crystalline) of uniformly bounded Hodge-Tate weights for all $n \geq 1$, then $T$ is also semi-stable (resp. crystalline). In this note, we show that we can relax the condition of "uniformly bounded Hodge-Tate weights" to an unbounded (log-)growth condition.
We prove that, on average, elliptic curves over Q have finitely many primes p for which they possess a p-adic point of order p. We include a discussion of applications to companion forms and the deformation theory of Galois representations.
We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze.
Let $ρ$ be a mod $\ell$ Galois representation. We show how to compute $ρ$, given the characteristic polynomial of the image of the Frobenius at one prime $p$ and a curve $C$ whose Jacobian contains $ρ$ in its $\ell$-torsion. The main ingredient is a method to $p$-adically lift torsion points on a Jacobian in the framework of Makdisi's algorithms.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$.
Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρ_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of ρ_E for a "random" elliptic curve E over k. In particular, if k\neq Q is linearly disjoint from the cyclotomic extension of Q, then ρ_E will be surjective for "most" elliptic curves over k.
We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some "generalized Hasse invariants" adapted to the Ekedahl-Oort stratification of the special fiber.
We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.
Let $F$ be a totally real number field and let $p$ be a prime unramified in $F$. We prove the existence of Galois pseudo-representations attached to mod $p^m$ Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties for $F$ of level prime to $p$.
To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and Rütsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explici…
Fix a positive integer $g$ and rational prime $p$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $p$ representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application, we consider the tame inverse Galois problem and are able to realise general symplectic groups as Galois groups of tame extensions of $\mathbb{Q}$.
We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic is…
Given a rational variety $V$ defined over $K$, we consider a principally polarized abelian variety $A$ of dimension $g$ defined over $V$. For each prime l we then consider the galois representation on the $l$-torsion of $A_t$, where $t$ is a $K$-rational point of $V$. The largest possible image is $GSp_{2g}(\mathbb{F}_l)$ and in the cases $g=1$ and 2, we are able to get surjectivity for all $l$ and almost all $t$. In the case $g=1$ this recovers a theorem originally proven by William Duke.
We describe the relations among the $\ell$-torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the $\ell$-torsion conjecture for $\ell$-groups and the other two conjectures for nilpotent groups.
Let K be a complete discrete valuation field of mixed characteristic (0,p) with perfect residue field. Let (π_n)_{n\ge 0} be a system of p-power roots of a uniformizer π=π_0 of K with π^p_{n+1}=π_n, and define G_s (resp.\ G_{\infty}) the absolute Galois group of K(π_s) (resp.\ K_{\infty}:=\bigcup_{n\ge 0} K(π_n)). In this paper, we study G_s-equivatiantness properties of G_{\infty}-equivariant homomorphisms between torsion (potentially) crystalline representations
In this paper we interpret the solutions to a particular Galois embedding problem over an extension K/F whose Galois group is a finite, cyclic p group in terms of certain Galois submodules within the parameterizing space of elementary p-abelian extensions of K; here p is a prime. Combined with some basic facts about the module structure of this parameterizing space, this allows us to exhibit a class of p-groups whose realization multiplicity is unbounded.
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$ without CM. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree recorded as a label of the edge. An isogeny-torsion graph is an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the correspond…
In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal $\fl$ of $\F_q[T]$, the question essentially asks whether, up to isogeny, a Drinfeld module $ϕ$ over $\F_q(T)$ contains a rational $\fl$-torsion point if the reduction of $ϕ$ at almost all primes of $\F_q[T]$ contains a rational $\fl$-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is $2$, but negative if the rank…
Typos in the abstract have been corrected. Let $ρ_n$ be an ordinary weight two representation of absolute Galois group of the rationals to $GL_2(\mathcal O/π^n)$. Here $\mathcal O$ is a ramified DVR with uniformiser $π$. If $ρ_n$ satisfies mild hypotheses we lift it to a characteristic zero $\mathcal O$-valued geometric weight two representation. The earlier methods could handle only the unramified case. We show that the deformation ring of a residual representation arising from a newform can be arranged to be a prescribed DVR provided we choose a suitable auxiliary level. We extend earlier pr…
We outline the proof of a theorem of M. Baker and A. Tamagawa that gives a complete description of the torsion points on a modular curve embedded in its Jacobian using the notion of an `almost rational torsion point.'
We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31.
What is the true order of growth of torsion in the cohomology of an arithmetic group? Let $D$ be a quaternion over an imaginary quadratic field $F.$ Let $E/F$ be a cyclic Galois extension with $\mathrm{Gal}(E/F) = \langle σ\rangle.$ We prove lower bounds for "the Lefschetz number of $σ$ acting on torsion cohomology" of certain Galois-stable arithmetic subgroups of $D_E^\times.$ For these same subgroups, we unconditionally prove a would-be-numerical consequence of the existence of a hypothetical base change map for torsion cohomology.
Given an elliptic curve $E$ over a global function field $K$, the Galois action on the $n$-torsion points of $E$ gives rise to a mod-n Galois representation $ρ_{E,n}$. For $K$ satisfying some mild conditions, we show that the set of $E$ for which $ρ_{E,n}$ is as large as possible for all $n$, has density $1$.
In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the …
We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Ruetsche, respectively. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.
Let $G$ be a finite group and let $N/E$ be a tamely ramified $G$-Galois extension of number fields. We show how Stickelberger's factorization of Gauss sums can be used to determine the stable isomorphism class of various arithmetic $\mathbb{Z}[G]$-modules attached to $N/E$. If $\mathcal{O}_N$ and $\mathcal{O}_E$ denote the rings of integers of $N$ and $E$ respectively, we get in particular that $\mathcal{O}_N\otimes_{\mathcal{O}_E}\mathcal{O}_N$ defines the trivial class in the class group $\mathrm{Cl}(\mathbb{Z}[G])$ and, if $N/E$ is also assumed to be locally abelian, that the square root of…
For any elliptic curve E defined over the rationals with complex multiplication and for every prime p, we describe the image of the mod p Galois representation attached to E. We deduce information about the field of definition of torsion points of these curves, in particular we classify all cases where there are torsion points over Galois number fields not containing the field of definition of the CM.
Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden.
We prove that the groups PSL(r,q^d) can be realized F_q(T)-regularly as Galois groups over the purely transcendental field F_q(T)(t_1,...,t_{r-1}) if r is even and r/2 is coprime to q^d-1. The method is to use twisted moduli schemes of Drinfeld modules (a la Shih). If one has sufficient control over modular forms, it can be made into an algorithm to find explicit equations. We do this for PSL(2,q^2) using Drinfeld modular forms.
Recently there has been much interest in studying the torsion subgroups of elliptic curves base-extended to infinite extensions of $\mathbb{Q}$. In this paper, given a finite group $G$, we study what happens with the torsion of an elliptic curve $E$ over $\mathbb{Q}$ when changing base to the compositum of all number fields with Galois group $G$. We do this by studying a group theoretic condition called generalized $G$-type, which is a necessary condition for a number field with Galois group $H$ to be contained in that compositum. In general, group theory allows one to reduce the original prob…
In this paper, we survey some Galois-theoretic techniques for studying torsion points on curves. In particular, we give new proofs of some results of A. Tamagawa and the present authors for studying torsion points on curves with "ordinary good" or "ordinary semistable" reduction at a given prime. We also give new proofs of: (1) The Manin-Mumford conjecture: There are only finitely many torsion points lying on a curve of genus at least 2 embedded in its Jacobian by an Albanese map; and (2) The Coleman-Kaskel-Ribet conjecture: If p is a prime number which is at least 23, then the only torsion po…
Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(D_4^\infty)$ be the compositum of all extensions of $\mathbb{Q}$ whose Galois closure has Galois group isomorphic to a quotient of a subdirect product of a finite number of transitive subgroups of $D_4$. In this article we first show that $\mathbb{Q}(D_4^\infty)$ is in fact the compositum of all $D_4$ extensions of $\mathbb{Q}$ and then we prove that the torsion subgroup of $E(\mathbb{Q}(D_4^\infty))$ is finite and determine the 24 possibilities for its structure. We also give a complete classification of the elliptic curves that have…
In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and Tamagawa, related with the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion.
In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.
For a $p$-adic local field $K$ with perfect residue field, L. Herr constructed a complex which computes the Galois cohomology of a $p$-torsion representation of the absolute Galois group of $K$ by using the theory of $(Φ,Γ)$-modules. We shall generalize his work to the imperfect residue field (the residue field has a finite $p$-basis) case.
We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.
Let $F$ be a local non-Archimedean field with ring of integers $o$. Let $\bf X$ be a one-dimensional formal $o$-module of $F$-height $n$ over the algebraic closure of the residue field of $o$. By the work of Drinfeld, the universal deformation $X$ of $\bf X$ is a formal group over a power series ring $R_0$ in $n-1$ variables over the completion of the maximal unramified extension of $o$. For $h \in \{0,...,n-1\}$ let $U_h$ be the subscheme of $\Spec(R_0)$ where the connected part of the associated divisible module of $X$ has height $h$. Using the theory of Drinfeld level structures we show tha…
For $\ell = 3$ and 5 it is known that every odd, irreducible, 2-dimensional representation of $\Gal(\bar{\Q}/\Q)$ with values in $\F_\ell$ and determinant equal to the cyclotomic character must "come from" the $\ell$-torsion points of an elliptic curve defined over $\Q$. We prove, by giving concrete counter-examples, that this result is false for every prime $\ell >5$.
Let E/Q be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of Q obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group H^1(G, E[p]) does not vanish, and investigate the analogous question for E[p^i] when i>1. We include an application to the verification of certain cases of the Birch and Swinnerton-Dyer conjecture, and another application to the Grunwald-Wang problem for elliptic curves.
Let K be a fixed number field and G its absolute Galois group. We give a bound C(K), depending only on the degree, the class number and the discriminant of K, such that for any elliptic curve E defined over K and any prime number p strictly larger than C(K), the image of the representation of G attached to the p-torsion points of E contains a subgroup of homotheties of index smaller than 12.
In this paper we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic $\ell>3$ as the Galois group of a tamely ramified Galois extension of $\mathbb{Q}$. The strategy is to consider the Galois representation $ρ_{\ell}$ attached to the Tate module at $\ell$ of a suitable abelian surface. We need to choose the abelian varieties carefully in order to ensure that the image of $ρ_{\ell}$ is large and simultaneously maintain a control on the ramification of the corresponding Galois extension. We obtain an explicit family of curves of genus 2 such tha…
In this short note we provide a few examples of non-isomorphic arithmetically equivalent global function fields. These examples are obtained via well-known technique of adjoining the torsion points of various Drinfeld Modules to realise the $Gl_n(\mathbb F_q)$ as a Galois group of extensions of global function fields. Furthermore we afford the code of the Magma scripts to verify the results and construct more examples in similar fashion.
Let K be a number field and let $\mathcal{E}$ be an elliptic curve defined over $K$. Let $m$ be a positive integer. We denote by $K(\mathcal{E}[m])$ the number fields obtained by adding to $K$ the coordinates of the $m$-torsion points of $\mathcal{E}$. We look for small (sometimes "minimal") set of generators of $K(\mathcal{E}[m])$. For $m=3$ and $m=4$, we describe explicit generators, degree and Galois groups of the extensions $K(\mathcal{E}[m])/K$.
Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's quadratic twists.
We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
We consider a dynamical system consisting of a pair of commuting power series, one noninvertible and another nontorsion invertible, of height one with coefficients in the $p$-adic integers. Assuming that each point of the dynamical system generates a Galois extension over the base field, we show that these extensions are in fact abelian, and, using results and considerations from the theory of the field of norms, we also show that the dynamical system must include a torsion series of maximal order. From an earlier result, this shows that the series must in fact be endomorphisms of some height-…
In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].
Let A be an abelian variety of positive dimension defined over a number field K and let Kbar be a fixed algebraic closure of K. For each element sigma of the absolute Galois group Gal(Kbar/K), let Kbar(sigma) be the fixed field of sigma in Kbar. We shall prove that the torsion subgroup of A(Kbar(sigma)) is infinite for all sigma in Gal(Kbar/K) outside of some set of Haar measure zero. This proves the number field case of a conjecture of Geyer and Jarden from 1978.
